1 Introduction
2 Data and Methods
2.1 The Data
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‘Finally, which are the two categories that get the highest benefits from Erasmus + mobility? (Please, click the first and the second category of possible recipients)’. The option was given to specify a first and second selection out of five possible beneficiaries: students/apprentices, schools and training centres, companies (both sending and hosting), the labour market, and the European Union as an institution. The beneficiaries were ranked on a three-level ordinal scale.The question posed only to the schools and companies was:
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‘Finally, which are the categories of possible recipients that get the highest benefits and the ones that get the lowest ones from Erasmus + mobility? Please, order the categories from 1 (highest) to 5 (lowest benefits)’. The same categories presented to the participants were used here. The beneficiaries were ranked on a five-level ordinal scale.
2.2 Models and Methods
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We can use just the top position of each distribution. Thus, the score estimates are proportional to the frequencies of the top positions obtained by the \(A\) alternatives. This estimation method uses basic information and does not require particular statistical techniques. In the following, this procedure is called ‘first position’.
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We can deduce scores from the averages of frequency distributions. This estimation method uses the full frequency distribution as if each ranking was an empirical occurrence of a random variable distributed with mean \(\left( {A + 1} \right)/2\) and variance \(\left( {A^{2} - 1} \right)/12\). Henceforth, we will call this method the ‘univariate’ procedure.
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We can model the relations between the \(A\left( {A - 1} \right)/2\) distinct pairs of alternatives by comparing their ranking positions and defining the level of ‘dominance’ of alternative \(a\) over alternative \(b\). Thus, this weighting procedure is called ‘bivariate’. This estimation method requires the construction of a ‘dominance matrix’, which reflects the bivariate relationships between alternatives. The way such a matrix is defined and how it can be processed for weight estimation purposes is described later in this section.
2.3 Criteria for Choosing the Best Approach
3 An Application to Mobility Beneficiaries
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The difference in estimates using just the first positions with respect to the methods based on the full distribution of evaluations is notable: the average of the participants’ scores computed over all assessors was almost 80%, while the average of all other methods was 36.9% and the median was 36.7%. This suggests that the estimates based on just the first position of the rankings should be considered a shortcut, as it is not influenced by the relations between beneficiaries. Since the estimation procedure based on just the first positions is an outlier among the applied models, it will not be considered further here.
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Among the other computational methods, Model 2 rules were computationally the least troublesome. The difference between the most extreme score-profiles obtained by adopting the quadratic distance of alternatives and that obtained applying the absolute distance with the same RS rule is relevant: the quadratic strategy produced scores that were, averaging the benefit perceived by all assessors for participants as beneficiaries, more than 10% larger (42.4% instead of 31.6%) than those obtained with the absolute weighting rule. The scores assigned to the other beneficiaries were reduced proportionally to the score assigned to the first beneficiary but were consistent with each other. The score assigned to the top beneficiary according to the ROC rule follows immediately that of the RS with quadratic weighting rule (40.9% on average), then the RR (36.7%) and the SR (34.1) scores. These results are consistent with the literature.
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The weighting rules seemed not so important if scores were estimated through the analysis of dominances. The quadratic procedure yielded results similar to the linearly-weighted analysis of dominances: the mean difference was lower than 0.6% whichever assessing-group was considered. Furthermore, the differences between assessors using either linear or quadratic powers for weight estimation were less important than those obtained in the univariate analysis. The application of any weighting rule to the analysis of dominances (Model 4) led to score profiles that did not diverge with each other. Actually, the first-beneficiary scores maintained the ordering highlighted in case of univariate analysis (Model 2): the largest value was obtained by the quadratic weighting (37.0% on average), ROC (36.9%), RR (36.8%), SR (36.7%), and RS (36.7%) rules.
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We ascertained just three cases of equal scores between alternatives: once when mediating the scores obtained by three groups of assessors with Model 3 and twice when applying linear weights to Model 4. This may mean that even a flatting rule applied together with the analysis of dominances tends to assign to alternative scores that differ from each other. This result is in line with the literature (e.g.Landau 1951; Hemelrijk, Wantia and Gygax 2005).
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Whatever the weighting rule, there was a clear hierarchy that placed participants at the top of the ranking with an endorsement of approximately 37%, then schools and companies at approximately 19% each, and finally, the labour market and the EU as an institution with approximately 12% each.
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The estimates of the distance between the collected rankings and the reference ranking did not differentiate among models because the reference ranking of the single models was about the same for all models. Thus, the measure of distance for Models 2 and 3 were the same for every group of assessors, but only for the RR rule. Small differences, instead, were found between Models 3 and 4 regardless of the weighting rule. The relative invariance of the distances (no difference exceeded 2.5%) was caused by both the near-linearity of the assessed alternatives and the shortness of the ranking.
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Models 3 and 4, which were based on the analysis of dominances, were able to produce biased estimates of the scores in a lower proportion than Model 2, which was based on just the univariate analysis of frequencies.
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Model 3, which was based on an equal-weight system, performed better in terms of variability than Model 4, for which we introduced various articulated weighting systems, but its bias was higher than any of the weighting systems in Model 4.7
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All weighting rules applied to dominance analysis showed the same level of variability. Instead, the bias slightly differed among the applied rules: the ROC and RR rules showed the lowest bias, while the SR and the RS with linear weighting rules showed an 8% higher bias. The highest bias (almost twice that of the ROC rule) was found for the RS rule with quadratic weighting.
4 Discussion and Conclusion
Models by assessor and weighting power | Beneficiaries of mobility (Alternatives) | ||||
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Participants | Schools and training c. | Companies (send/host) | Labour market | EU as an institution | |
1—Participants | 69.7 | 11.0 | 9.3 | 3.6 | 6.4 |
1—Schools | 90.7 | 2.3 | 2.3 | 0.0 | 4.7 |
1—Companies | 75.8 | 6.4 | 8.4 | 2.0 | 7.4 |
1—Mean of Assessor Groups | 78.7 | 6.6 | 6.7 | 1.9 | 6.1 |
2—Participants, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 31.6 | 18.3 | 18.5 | 15.5 | 16.1 |
2—Participants, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 43.9 | 16.3 | 16.5 | 11.0 | 12.3 |
2—Schools, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 32.8 | 21.3 | 18.3 | 15.1 | 12.5 |
2—Schools, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 43.8 | 20.3 | 15.8 | 11.2 | 8.9 |
2—Companies, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 30.5 | 19.1 | 21.6 | 15.2 | 13.6 |
2—Companies, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 39.5 | 17.3 | 21.3 | 11.5 | 10.4 |
2—Assessors Average, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 31.6 | 19.6 | 19.5 | 15.2 | 14.1 |
2—Assessors Average, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 42.4 | 18.0 | 17.9 | 11.2 | 10.5 |
2—Participants ROC | 39.5 | 16.9 | 17.0 | 12.8 | 13.8 |
2—Schools ROC | 43.8 | 18.9 | 15.5 | 11.7 | 10.1 |
2—Companies ROC | 39.4 | 16.9 | 20.2 | 12.1 | 11.4 |
2—Assessors Average ROC | 40.9 | 17.6 | 17.5 | 12.2 | 11.8 |
2—Participants RR | 32.8 | 17.9 | 17.8 | 15.4 | 16.1 |
2—Schools RR | 39.8 | 20.0 | 16.6 | 12.0 | 11.6 |
2—Companies RR | 37.8 | 16.6 | 18.9 | 13.1 | 13.6 |
2—Assessors Average RR | 36.7 | 18.2 | 17.8 | 13.5 | 13.8 |
2—Participants SR | 32.2 | 18.1 | 18.2 | 15.4 | 16.1 |
2—Schools SR | 36.7 | 19.7 | 17.0 | 14.1 | 12.5 |
2—Companies SR | 33.6 | 18.0 | 20.5 | 14.3 | 13.6 |
2—Assessors Average SR | 34.1 | 18.6 | 18.6 | 14.6 | 14.1 |
3—Participants | 33.7 | 19.0 | 19.7 | 13.1 | 14.5 |
3—Schools | 39.7 | 20.2 | 16.7 | 12.7 | 10.7 |
3—Companies | 35.0 | 18.3 | 21.3 | 13.3 | 12.1 |
3—Average of Assessors | 36.1 | 19.2 | 19.2 | 13.1 | 12.4 |
3—All assessors, P matrix | 35.8 | 19.2 | 19.3 | 13.2 | 12.5 |
4—Participants, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 34.4 | 19.2 | 19.2 | 12.7 | 14.5 |
4—Participants, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 34.8 | 19.4 | 19.0 | 12.1 | 14.7 |
4—Schools, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 39.9 | 20.2 | 16.7 | 12.2 | 11.0 |
4—Schools, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 39.9 | 20.1 | 16.7 | 11.7 | 11.6 |
4—Companies, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 35.8 | 18.0 | 21.2 | 12.5 | 12.5 |
4—Companies, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 36.2 | 17.9 | 21.1 | 11.8 | 13.0 |
4—Assessors Average, RS \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 36.7 | 19.1 | 19.0 | 12.5 | 12.7 |
4—Assessors Average, RS \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 37.0 | 19.1 | 18.9 | 11.9 | 13.1 |
4—Participants ROC | 34.7 | 19.3 | 19.1 | 12.3 | 14.6 |
4—Schools ROC | 39.8 | 20.1 | 16.7 | 11.8 | 11.6 |
4—Companies ROC | 36.1 | 17.8 | 21.0 | 12.0 | 13.1 |
4—Assessors Average ROC | 36.9 | 19.1 | 18.9 | 12.0 | 13.1 |
4—Participants RR | 34.4 | 19.2 | 19.2 | 12.7 | 14.5 |
4—Schools RR | 39.8 | 20.0 | 16.6 | 12.0 | 11.6 |
4—Companies RR | 36.0 | 17.8 | 20.9 | 12.3 | 13.0 |
4—Assessors Average RR | 36.8 | 19.0 | 18.9 | 12.3 | 13.0 |
4—Participants SR | 34.4 | 19.2 | 19.2 | 12.7 | 14.5 |
4—Schools SR | 39.9 | 20.1 | 16.7 | 12.1 | 11.2 |
4—Companies SR | 35.9 | 17.9 | 21.1 | 12.4 | 12.7 |
4—Assessors Average SR | 36.7 | 19.1 | 19.0 | 12.4 | 12.8 |
Mean Score: Participantsa | 35.1 | 18.4 | 18.5 | 13.2 | 14.7 |
Mean Score: Schoolsa | 39.6 | 20.1 | 16.7 | 12.4 | 11.2 |
Mean score: Companiesa | 36.0 | 17.8 | 20.8 | 12.8 | 12.6 |
Mean Score: Overalla | 36.9 | 18.8 | 18.7 | 12.8 | 12.8 |
Model | Participants | Schools | Companies | Overall | ||||
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Distance | Bias | Distance | Bias | Distance | Bias | Distance | Bias | |
2, \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 0.538 | 0.0147 | 0.294 | 0.0273 | 0.337 | 0.0219 | 0.498 | 0.0212 |
2, \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 0.538 | 0.0351 | 0.294 | 0.0176 | 0.337 | 0.0160 | 0.498 | 0.0220 |
2, ROC | 0.538 | 0.0175 | 0.294 | 0.0167 | 0.337 | 0.0137 | 0.498 | 0.0160 |
2, RR | 0.558 | 0.0142 | 0.294 | 0.0023 | 0.337 | 0.0124 | 0.510 | 0.0065 |
2, SR | 0.538 | 0.0142 | 0.294 | 0.0132 | 0.378 | 0.0108 | 0.498 | 0.0121 |
3 | 0.538 | 0.0071 | 0.294 | 0.0020 | 0.337 | 0.0061 | 0.484 | 0.0049 |
4, \(\delta \hspace{0.17em}=\hspace{0.17em}1\) | 0.538 | 0.0059 | 0.294 | 0.0017 | 0.337 | 0.0024 | 0.510 | 0.0027 |
4, \(\delta \hspace{0.17em}=\hspace{0.17em}2\) | 0.538 | 0.0059 | 0.294 | 0.0029 | 0.378 | 0.0039 | 0.510 | 0.0037 |
4, ROC | 0.558 | 0.0059 | 0.294 | 0.0025 | 0.378 | 0.0031 | 0.510 | 0.0025 |
4, RR | 0.538 | 0.0059 | 0.294 | 0.0023 | 0.378 | 0.0019 | 0.510 | 0.0025 |
4, SR | 0.538 | 0.0059 | 0.294 | 0.0013 | 0.378 | 0.0018 | 0.510 | 0.0027 |